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On Triangular Lucas Numbers

  • Ming Luo

Abstract

In the paper [3], we have proved that the only triangular numbers (i.e., the positive integers of the form \( \frac{1}{2}m \)(m+1)) in the Fibonacci sequence
$$ {u_n} + 2 = {u_{n + 1}} + {u_{{n^,}}}{u_0} = 0, {u_1} = 1 $$
are u ±1=u2=1, u4=3, u8=21 and u10=55. This verifies a conjecture of Vern Hoggatt [2]. In this paper we shall find all triangular numbers in the Lucas sequence
$$ {v_n} + 2 = {v_{n + 1}} + {v_{{n^,}}}{v_0} = 2, {v_1} = 1, $$
where n ranges over all integers.

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References

  1. [1]
    Cohn, J. H. E. “On Square Fibonacci Numbers.” J. London Math. Soc. 39 (1964): pp. 537–541.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Guy, R. K. Unsolved Problems in Number Theory. New York: Springer-Verlag, 1981, p. 106.zbMATHGoogle Scholar
  3. [3]
    Luo, Ming. “On Triangular Fibonacci Numbers.” The Fibonacci Quarterly, 27.2 (1989): pp. 98–108.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1991

Authors and Affiliations

  • Ming Luo

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